Estimating Control Points for B-Spline Surfaces Using Fully Populated Synthetic Variance–Covariance Matrices for TLS Point Clouds

Abstract

A flexible approach for geometric modelling of point clouds obtained from Terrestrial Laser Scanning (TLS) is by means of B-splines. These functions have gained some popularity in the engineering geodesy as they provide a suitable basis for a spatially continuous and parametric deformation analysis. In the predominant studies on geometric modelling of point clouds by B-splines, uncorrelated and equally weighted measurements are assumed. Trying to overcome this, the elementary errors theory is applied for establishing fully populated covariance matrices of TLS observations that consider correlations in the observed point clouds. In this article, a systematic approach for establishing realistic synthetic variance–covariance matrices (SVCMs) is presented and afterward used to model TLS point clouds by B-splines. Additionally, three criteria are selected to analyze the impact of different SVCMs on the functional and stochastic components of the estimation results. Plausible levels for variances and covariances are obtained using a test specimen of several dm—dimension. It is used to identify the most dominant elementary errors under laboratory conditions. Starting values for the variance level are obtained from a TLS calibration. The impact of SVCMs with different structures and different numeric values are comparatively investigated. Main findings of the paper are that for the analyzed object size and distances, the structure of the covariance matrix does not significantly affect the location of the estimated surface control points, but their precision in terms of the corresponding standard deviations. Regarding the latter, properly setting the main diagonal terms of the SVCM is of superordinate importance compared to setting the off-diagonal ones. The investigation of some individual errors revealed that the influence of their standard deviation on the precision of the estimated parameters is primarily dependent on the scanning distance. When the distance stays the same, one-sided influences on the precision of the estimated control points can be observed with an increase in the standard deviations.

Two-epoch TLS deformation analysis of a double curved wooden structure using approximating B-spline surfaces and fully-populated synthetic covariance matrices

“…The workflow is applied to a part of a double curved 14 m high wooden structure, called the Urbach Tower. This approach continues the work of Raschhofer et al (2021) by extending the analysis to object dimension of several meters as well as to measurement distances of tenth of meters. This implies including the atmospheric conditions in the SVCM, applying a different instrumental error model and using a different source of reference data.…”

“…Since this surface is only used as reference surface and no functional correlating errors are known for the Surphaser scanner, the identity matrix is used here. As seen in Raschhofer et al (2021), the differences of estimated control points position for an estimated surface at close range (section 4.1. in aforementioned publication) using identity, diagonal or fully populated matrix as a stochastic model are negligible. Since no stochastic information is needed for the projection, the identity matrix is chosen for simplicity.…”

“…As for the stochastic model, a SVCM is introduced into the estimation. For each point cloud, the SVCM is adjusted by varying the standard deviations of the noncorrelating errors, as performed in Raschhofer et al, (2021), so that the null hypothesis of the global test of the individual B-spline surface estimation cannot rejected. Table 2 shows the values obtained for the standard deviations of the non-correlating errors.…”

“…The stochastic model used to describe the TLS observations' uncertainty is established based on the elementary error theory. Specifically, the model is represented by a synthetic variance-covariance matrix (SVCM) constructed as explained inKauker and Schwieger (2017),Kerekes and Schwieger (2020) andRaschhofer et al (2021). Without resuming the theory of defining fully populated VCM 𝛴 in observation space or the equivalent 𝛴 in coordinate space, only5 th Joint International Symposium on Deformation Monitoring (JISDM), 20-22 June 2022, Valencia, Spain 2022, Editorial Universitat Politècnica de València the relevant different aspects to the aforementioned publications are highlighted.…”

Classification of Terrestrial Laser Scanner Point Clouds: A Comparison of Methods for Landslide Monitoring from Mathematical Surface Approximation

“…[13] for bathymetry data set, or [14] for turbine blade design, to cite but a few). The main advantages of mathematical surfaces is to reduce the point clouds to a few parameters while simultaneously allowing spatially continuous and parametric deformation analysis [15,16]. However, the accuracy and smoothness of surface approximations are affected by uneven objects, such as trees, low vegetation, stones, houses, and roads.…”